This method allows you to simulate vector-valued Hull-White/Vasicek
processes of the form:

dXt=S(t)[L(t)−Xt]dt+V(t)dWt

where:

Xt is an NVARS-by-1 state
vector of process variables.

S is an NVARS-by-NVARS of
mean reversion speeds (the rate of mean reversion).

L is an NVARS-by-1 vector
of mean reversion levels (long-run mean or level).

V is an NVARS-by-NBROWNS instantaneous
volatility rate matrix.

dWtis an NBROWNS-by-1 Brownian
motion vector.

Input Arguments

Specify required input parameters as one of the following types:

A MATLAB® array. Specifying an array indicates
a static (non-time-varying) parametric specification. This array fully
captures all implementation details, which are clearly associated
with a parametric form.

A MATLAB function. Specifying a function provides
indirect support for virtually any static, dynamic, linear, or nonlinear
model. This parameter is supported via an interface, because all implementation
details are hidden and fully encapsulated by the function.

Note

You can specify combinations of array and function input parameters
as needed.

Moreover, a parameter is identified as a deterministic function
of time if the function accepts a scalar time t as
its only input argument. Otherwise, a parameter is assumed to be a
function of time t and state X(t) and
is invoked with both input arguments.

The required input parameters are:

Speed

Speed represents the function S.
If you specify Speed as an array, it must be an NVARS-by-NVARS matrix
of mean-reversion speeds (the rate at which the state vector reverts
to its long-run average Level). As a deterministic
function of time, when Speed is called with a real-valued
scalar time t as its only input, Speed must
produce an NVARS-by-NVARS matrix.
If you specify Speed as a function of time and
state, it calculates the speed of mean reversion. This function must
generate an NVARS-by-NVARS matrix
of reversion rates when called with two inputs:

A real-valued scalar observation time t.

An NVARS-by-1 state
vector Xt.

Level

Level represents the function L.
If you specify Level as an array, it must be an NVARS-by-1 column
vector of reversion levels. As a deterministic function of time, when Level is
called with a real-valued scalar time t as its
only input, Level must produce an NVARS-by-1 column
vector. If you specify Level as a function of time
and state, it must generate an NVARS-by-1 column
vector of reversion levels when called with two inputs:

A real-valued scalar observation time t.

An NVARS-by-1 state
vector Xt.

Sigma

Sigma represents the parameter V.
If you specify Sigma as an array, it must be an NVARS-by-NBROWNS matrix
of instantaneous volatility rates. In this case, each row of Sigma corresponds
to a particular state variable. Each column corresponds to a particular
Brownian source of uncertainty, and associates the magnitude of the
exposure of state variables with sources of uncertainty. As a deterministic
function of time, when Sigma is called with a real-valued
scalar time t as its only input, Sigma must
produce an NVARS-by-NBROWNS matrix.
If you specify it as a function of time and state, Sigma must
return an NVARS-by-NBROWNS matrix
of volatility rates when invoked with two inputs:

A real-valued scalar observation time t.

An NVARS-by-1 state
vector Xt.

Note

Although the constructor does not enforce restrictions on the
signs of any of these input arguments, each argument is specified
as a positive value.

Specify the parameter name as a character vector,
followed by its corresponding parameter value.

You can specify parameter name/value pairs in any
order.

Parameter names are case insensitive.

You can specify unambiguous partial character vector
matches.

Valid parameter names are:

StartTime

Scalar starting time of the first observation, applied to all
state variables. If you do not specify a value for StartTime,
the default is 0.

StartState

Scalar, NVARS-by-1 column
vector, or NVARS-by-NTRIALS matrix
of initial values of the state variables.

If StartState is
a scalar, hwv applies the same initial value to
all state variables on all trials.

If StartState is
a column vector, hwv applies a unique initial value
to each state variable on all trials.

If StartState is
a matrix, hwv applies a unique initial value to
each state variable on each trial.

If you do not specify
a value for StartState, all variables start at 1.

Correlation

Correlation between Gaussian random variates drawn to
generate the Brownian motion vector (Wiener processes). Specify Correlation as
an NBROWNS-by-NBROWNS positive
semidefinite matrix, or as a deterministic function C(t) that
accepts the current time t and returns an NBROWNS-by-NBROWNS positive
semidefinite correlation matrix.

A Correlation matrix
represents a static condition.

As a deterministic function
of time, Correlation allows you to specify a dynamic
correlation structure.

If you do not specify a value for Correlation,
the default is an NBROWNS-by-NBROWNS identity
matrix representing independent Gaussian processes.

Simulation

A user-defined simulation function or SDE simulation method.
If you do not specify a value for Simulation, the
default method is simulation by Euler approximation (simByEuler).

Output Arguments

HWV

Object of class hwv with the following
displayed parameters:

StartTime: Initial observation
time

StartState: Initial state at StartTime

Correlation: Access function for
the Correlation input, callable as a function of
time

Drift: Composite drift-rate function,
callable as a function of time and state

Diffusion: Composite diffusion-rate
function, callable as a function of time and state

Simulation: A simulation function
or method

Speed: Access function for the
input argument Speed, callable as a function of
time and state

Level: Access function for the
input argument Level, callable as a function of
time and state

Sigma: Access function for the
input argument Sigma, callable as a function of
time and state

Examples

Algorithms

When you specify the required input parameters as arrays, they
are associated with a specific parametric form. By contrast, when
you specify either required input parameter as a function, you can
customize virtually any specification.

Accessing the output parameters with no inputs simply returns
the original input specification. Thus, when you invoke these parameters
with no inputs, they behave like simple properties and allow you to
test the data type (double vs. function, or equivalently, static vs.
dynamic) of the original input specification. This is useful for validating
and designing methods.

When you invoke these parameters with inputs, they behave like
functions, giving the impression of dynamic behavior. The parameters
accept the observation time t and a state vector Xt,
and return an array of appropriate dimension. Even if you originally
specified an input as an array, hwv treats it as
a static function of time and state, by that means guaranteeing that
all parameters are accessible by the same interface.